ó Ąź™\c@ s ddlmZmZddlmZmZmZmZddlm Z ddl m Z m Z m Z ddlmZmZddlmZddlmZmZddlmZdd lmZdd lmZdd lmZd efd „ƒYZdefd„ƒYZ defd„ƒYZ!defd„ƒYZ"defd„ƒYZ#defd„ƒYZ$defd„ƒYZ%e$Z&e%Z'defd„ƒYZ(dS(i˙˙˙˙(tprint_functiontdivision(tStsympifytDummytMod(tcacheit(treducetrangetHAS_GMPY(tFunctiontArgumentIndexError(t fuzzy_and(tIntegertpi(tEq(tsieve(tPoly(tsqrttCombinatorialFunctioncB seZdZd„ZRS(s(Base class for combinatorial functions. cC s@ddlm}||ƒ}||ƒ|||ƒkr<|S|S(Ni˙˙˙˙(tcombsimp(tsympy.simplify.combsimpR(tselftratiotmeasuretrationaltinverseRtexpr((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_simplifys  (t__name__t __module__t__doc__R(((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRst factorialc!B seZdZdd„Zdddddddddddd d d d d d ddddddddddddddddg!ZgZed„ƒZed„ƒZed„ƒZ d „Z d!„Z d"„Z d#„Z d$„Zd%„Zd&„Zd'„Zd(„ZRS()sŁImplementation of factorial function over nonnegative integers. By convention (consistent with the gamma function and the binomial coefficients), factorial of a negative integer is complex infinity. The factorial is very important in combinatorics where it gives the number of ways in which `n` objects can be permuted. It also arises in calculus, probability, number theory, etc. There is strict relation of factorial with gamma function. In fact `n! = gamma(n+1)` for nonnegative integers. Rewrite of this kind is very useful in case of combinatorial simplification. Computation of the factorial is done using two algorithms. For small arguments a precomputed look up table is used. However for bigger input algorithm Prime-Swing is used. It is the fastest algorithm known and computes `n!` via prime factorization of special class of numbers, called here the 'Swing Numbers'. Examples ======== >>> from sympy import Symbol, factorial, S >>> n = Symbol('n', integer=True) >>> factorial(0) 1 >>> factorial(7) 5040 >>> factorial(-2) zoo >>> factorial(n) factorial(n) >>> factorial(2*n) factorial(2*n) >>> factorial(S(1)/2) factorial(1/2) See Also ======== factorial2, RisingFactorial, FallingFactorial icC sbddlm}m}|dkrO||jddƒ|d|jddƒSt||ƒ‚dS(Ni˙˙˙˙(tgammat polygammaii(tsympyR!R"targsR (RtargindexR!R"((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pytfdiffSs -iiii#i;i?iľiçiť i­i#iSŤi{/i!†im´ińĚisXiUňiÇP ioăikÖiIi/„LiSůŞi}î“i#áéc C so|dkr|j|Stt|ƒƒg}}x‹tjd|dƒD]s}d|}}xAtr||}|dkr™|d@dkrš||9}qšq]Pq]W|dkrG|j|ƒqGqGWxJtj|d|ddƒD]*}||d@dkrÝ|j|ƒqÝqÝWd}}x0tj|dd|dƒD]}||9}q4Wx|D]}||9}qOW||SdS(Ni!iiii(t _small_swingtintt_sqrtRt primerangetTruetappend( tclstntNtprimestprimetptqt L_productt R_product((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_swingbs,       % % cC s6|dkrdS|j|dƒd|j|ƒSdS(Nii(t _recursiveR6(R-R.((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR7†s cC s5t|ƒ}|jr1|tjkr+tjS|tjkrAtjS|jr1|jrZtjS|j }|dkrÉ|j sľd}x4t ddƒD] }||9}|j j |ƒqŽWn|j |d}nXt rńddlm}|j|ƒ}n0t|ƒjdƒ}|j|ƒd||}t|ƒSq1ndS(Niii˙˙˙˙(tgmpyt1i(Rt is_NumberRtZerotOnetInfinityt is_Integert is_negativetComplexInfinityR2t_small_factorialsRR,R tsympy.core.compatibilityR8tfactbintcountR7R (R-R.tresulttiR8tbits((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pytevals.        c C s6dtt|ƒƒ}}dg|}d}xžtjd|dƒD]†}|dkrŠd||}}x!|r†||7}||}qiWn||krŻ||||||tmapR(RRRJR (RR3R.taqtdtisprimetfc((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt _eval_ModÍs*      !  cK sddlm}||dƒS(Ni˙˙˙˙(R!i(R#R!(RR.tkwargsR!((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_rewrite_as_gammaëscK sNddlm}|jrJ|jrJtddtƒ}|||d|fƒSdS(Ni˙˙˙˙(tProductRGtintegeri(R#RaRTRSRR+(RR.R_RaRG((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_rewrite_as_ProductďscC s(|jdjr$|jdjr$tSdS(Ni(R$RSRTR+(R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_is_integerős cC s(|jdjr$|jdjr$tSdS(Ni(R$RSRTR+(R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_is_positiveůs cC s.|jd}|jr*|jr*|djSdS(Nii(R$RSRT(Rtx((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt _eval_is_evenýs cC s.|jd}|jr*|jr*|djSdS(Nii(R$RSRT(RRf((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_is_composites cC s'|jd}|js|jr#tSdS(Ni(R$RTt is_nonintegerR+(RRf((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt _eval_is_reals (RRRR&R'RAt classmethodR6R7RIRRR^R`RcRdReRgRhRj(((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR "s$/ 6$"        tMultiFactorialcB seZRS((RR(((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRl st subfactorialcB s_eZdZeed„ƒƒZed„ƒZd„Zd„Zd„Z d„Z d„Z RS(sThe subfactorial counts the derangements of n items and is defined for non-negative integers as: .. math:: !n = \begin{cases} 1 & n = 0 \\ 0 & n = 1 \\ (n-1)(!(n-1) + !(n-2)) & n > 1 \end{cases} It can also be written as ``int(round(n!/exp(1)))`` but the recursive definition with caching is implemented for this function. An interesting analytic expression is the following [2]_ .. math:: !x = \Gamma(x + 1, -1)/e which is valid for non-negative integers `x`. The above formula is not very useful incase of non-integers. :math:`\Gamma(x + 1, -1)` is single-valued only for integral arguments `x`, elsewhere on the positive real axis it has an infinite number of branches none of which are real. References ========== .. [1] https://en.wikipedia.org/wiki/Subfactorial .. [2] http://mathworld.wolfram.com/Subfactorial.html Examples ======== >>> from sympy import subfactorial >>> from sympy.abc import n >>> subfactorial(n + 1) subfactorial(n + 1) >>> subfactorial(5) 44 See Also ======== sympy.functions.combinatorial.factorials.factorial, sympy.utilities.iterables.generate_derangements, sympy.functions.special.gamma_functions.uppergamma cC sJ|s tjS|dkr tjS|d|j|dƒ|j|dƒS(Nii(RR<R;t_eval(RR.((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRn;s  cC s[|jrW|jr(|jr(|j|ƒS|tjkr>tjS|tjkrWtjSndS(N(R:R>RTRnRtNaNR=(R-targ((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRIDs  cC s(|jdjr$|jdjr$tSdS(Ni(R$tis_oddRTR+(R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRgNs cC s(|jdjr$|jdjr$tSdS(Ni(R$RSRTR+(R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRdRs cK s(ddlm}||ddƒtjS(Ni˙˙˙˙(t uppergammai(R#RrRtExp1(RRpR_Rr((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_rewrite_as_uppergammaVscC s(|jdjr$|jdjr$tSdS(Ni(R$RSRTR+(R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_is_nonnegativeZs cC s(|jdjr$|jdjr$tSdS(Ni(R$tis_evenRTR+(R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt _eval_is_odd^s ( RRRRkRRnRIRgRdRtRuRw(((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRms)    t factorial2cB sJeZdZed„ƒZd„Zd„Zd„Zd„Zd„Z RS(s3The double factorial `n!!`, not to be confused with `(n!)!` The double factorial is defined for nonnegative integers and for odd negative integers as: .. math:: n!! = \begin{cases} 1 & n = 0 \\ n(n-2)(n-4) \cdots 1 & n\ \text{positive odd} \\ n(n-2)(n-4) \cdots 2 & n\ \text{positive even} \\ (n+2)!!/(n+2) & n\ \text{negative odd} \end{cases} References ========== .. [1] https://en.wikipedia.org/wiki/Double_factorial Examples ======== >>> from sympy import factorial2, var >>> var('n') n >>> factorial2(n + 1) factorial2(n + 1) >>> factorial2(5) 15 >>> factorial2(-1) 1 >>> factorial2(-5) 1/3 See Also ======== factorial, RisingFactorial, FallingFactorial cC sĽ|jrĄ|js!tdƒ‚n|jrg|jrO|d}d|t|ƒSt|ƒt|dƒS|jr’|tj d|dt| ƒStdƒ‚ndS(Ns<argument must be nonnegative integer or negative odd integerii( R:R>t ValueErrorRTRvR RxRqRt NegativeOne(R-Rptk((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRIˆs      "cC sP|jd}|jrL|jr#tS|jrL|jr9tS|jrItSqLndS(Ni(R$RSRqRXRvt is_positiveR+tis_zero(RR.((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRg s      cC sB|jd}|jr>|djr'tS|jr>|djSndS(Niii(R$RSRTR+Rq(RR.((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRdŹs     cC sK|jd}|jr!|djS|jrG|jr7tS|jrGtSndS(Nii(R$RqRTRvR|RXR}R+(RR.((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRwśs      cC sF|jd}|jrB|djr'tS|jrB|ddjSndS(Niii(R$RSRTR+RqRv(RR.((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyReÂs     cK sddlm}m}m}d|d||ddƒ|dtt|dƒdƒf|dtƒtt|dƒdƒfƒS(Ni˙˙˙˙(R!t PiecewiseRiii(R#R!R~RRRR(RR.R_R!R~R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR`Ís;( RRRRkRIRgRdRwReR`(((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRxcs# tRisingFactorialcB sSeZdZed„ƒZd„Zd„Zd„Zd„Zd„Z d„Z RS(s Rising factorial (also called Pochhammer symbol) is a double valued function arising in concrete mathematics, hypergeometric functions and series expansions. It is defined by: .. math:: rf(x,k) = x \cdot (x+1) \cdots (x+k-1) where `x` can be arbitrary expression and `k` is an integer. For more information check "Concrete mathematics" by Graham, pp. 66 or visit http://mathworld.wolfram.com/RisingFactorial.html page. When `x` is a Poly instance of degree >= 1 with a single variable, `rf(x,k) = x(y) \cdot x(y+1) \cdots x(y+k-1)`, where `y` is the variable of `x`. This is as described in Peter Paule, "Greatest Factorial Factorization and Symbolic Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268, 1995. Examples ======== >>> from sympy import rf, symbols, factorial, ff, binomial, Poly >>> from sympy.abc import x >>> n, k = symbols('n k', integer=True) >>> rf(x, 0) 1 >>> rf(1, 5) 120 >>> rf(x, 5) == x*(1 + x)*(2 + x)*(3 + x)*(4 + x) True >>> rf(Poly(x**3, x), 2) Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain='ZZ') Rewrite >>> rf(x, k).rewrite(ff) FallingFactorial(k + x - 1, k) >>> rf(x, k).rewrite(binomial) binomial(k + x - 1, k)*factorial(k) >>> rf(n, k).rewrite(factorial) factorial(k + n - 1)/factorial(n - 1) See Also ======== factorial, factorial2, FallingFactorial References ========== .. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol c s'tˆƒ‰t|ƒ}ˆtjks6|tjkr=tjSˆtjkrVt|ƒS|jr#|tjkrutjS|jrLˆtjkr”tjSˆtj kr˝|j rłtj StjSq t ˆt ƒr!ˆj }t|ƒdkrötdƒ‚qIt‡fd†tdt|ƒƒdƒSq t‡fd†tdt|ƒƒdƒSq#ˆtjkrbtjSˆtj krxtjSt ˆt ƒręˆj }t|ƒdkrątdƒ‚q dt‡fd†tdtt|ƒƒdƒdƒSq#dt‡fd†tdtt|ƒƒdƒdƒSndS(Nis0rf only defined for polynomials on one generatorc s|ˆj|ƒjƒS(N(tshifttexpand(trRG(Rf(sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt*sic s |ˆ|S(N((R‚RG(Rf(sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRƒ.tc s|ˆj| ƒjƒS(N(R€R(R‚RG(Rf(sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRƒ=sc s |ˆ|S(N((R‚RG(Rf(sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRƒAs(RRRoR<R R>R;R|R=tNegativeInfinityRqt isinstanceRtgenstlenRyRRR(RU(R-RfR{R‡((RfsGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRIsH         'cK s(ddlm}|||ƒ||ƒS(Ni˙˙˙˙(R!(R#R!(RRfR{R_R!((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR`EscK st||d|ƒS(Ni(tFallingFactorial(RRfR{R_((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt!_eval_rewrite_as_FallingFactorialIscK s6|jr2|jr2t||dƒt|dƒSdS(Ni(RSR (RRfR{R_((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_rewrite_as_factorialLscK s,|jr(t|ƒt||d|ƒSdS(Ni(RSR tbinomial(RRfR{R_((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt_eval_rewrite_as_binomialPs cC s1t|jdj|jdj|jdjfƒS(Nii(R R$RSRT(R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRdTscC s9ddlj}|j|jdjƒ|jdjƒƒS(Ni˙˙˙˙ii(tsage.alltalltrising_factorialR$t_sage_(Rtsage((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR‘Xs( RRRRkRIR`RŠR‹RRdR‘(((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRŘs47     R‰cB sSeZdZed„ƒZd„Zd„Zd„Zd„Zd„Z d„Z RS(sF Falling factorial (related to rising factorial) is a double valued function arising in concrete mathematics, hypergeometric functions and series expansions. It is defined by .. math:: ff(x,k) = x \cdot (x-1) \cdots (x-k+1) where `x` can be arbitrary expression and `k` is an integer. For more information check "Concrete mathematics" by Graham, pp. 66 or visit http://mathworld.wolfram.com/FallingFactorial.html page. When `x` is a Poly instance of degree >= 1 with single variable, `ff(x,k) = x(y) \cdot x(y-1) \cdots x(y-k+1)`, where `y` is the variable of `x`. This is as described in Peter Paule, "Greatest Factorial Factorization and Symbolic Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268, 1995. >>> from sympy import ff, factorial, rf, gamma, polygamma, binomial, symbols, Poly >>> from sympy.abc import x, k >>> n, m = symbols('n m', integer=True) >>> ff(x, 0) 1 >>> ff(5, 5) 120 >>> ff(x, 5) == x*(x-1)*(x-2)*(x-3)*(x-4) True >>> ff(Poly(x**2, x), 2) Poly(x**4 - 2*x**3 + x**2, x, domain='ZZ') >>> ff(n, n) factorial(n) Rewrite >>> ff(x, k).rewrite(gamma) (-1)**k*gamma(k - x)/gamma(-x) >>> ff(x, k).rewrite(rf) RisingFactorial(-k + x + 1, k) >>> ff(x, m).rewrite(binomial) binomial(x, m)*factorial(m) >>> ff(n, m).rewrite(factorial) factorial(n)/factorial(-m + n) See Also ======== factorial, factorial2, RisingFactorial References ========== .. [1] http://mathworld.wolfram.com/FallingFactorial.html c s-tˆƒ‰t|ƒ}ˆtjks6|tjkr=tjS|jr\ˆ|kr\tˆƒS|jr)|tjkr{tjS|jrRˆtj krštj Sˆtj krĂ|j rštj Stj Sq&t ˆt ƒr'ˆj}t|ƒdkrütdƒ‚qOt‡fd†tdt|ƒƒdƒSq&t‡fd†tdt|ƒƒdƒSq)ˆtj krhtj Sˆtj kr~tj St ˆt ƒrđˆj}t|ƒdkrˇtdƒ‚q&dt‡fd†tdtt|ƒƒdƒdƒSq)dt‡fd†tdtt|ƒƒdƒdƒSndS( Nis0ff only defined for polynomials on one generatorc s|ˆj| ƒjƒS(N(R€R(R‚RG(Rf(sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRƒąsic s |ˆ|S(N((R‚RG(Rf(sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRƒľR„s0rf only defined for polynomials on one generatorc s|ˆj|ƒjƒS(N(R€R(R‚RG(Rf(sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRƒĂsc s |ˆ|S(N((R‚RG(Rf(sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRƒÇR„(RRRoRSR R>R;R<R|R=R…RqR†RR‡RˆRyRRR(RU(R-RfR{R‡((RfsGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRI•sH         'cK s1ddlm}d||||ƒ|| ƒS(Ni˙˙˙˙(R!(R#R!(RRfR{R_R!((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR`ĘscK st||d|ƒS(Ni(trf(RRfR{R_((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyt _eval_rewrite_as_RisingFactorialÎscK s$|jr t|ƒt||ƒSdS(N(RSR RŒ(RRfR{R_((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRŃs cK s.|jr*|jr*t|ƒt||ƒSdS(N(RSR (RRfR{R_((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR‹ŐscC s1t|jdj|jdj|jdjfƒS(Nii(R R$RSRT(R((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyRdŮscC s9ddlj}|j|jdjƒ|jdjƒƒS(Ni˙˙˙˙ii(RŽRtfalling_factorialR$R‘(RR’((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR‘Ýs( RRRRkRIR`R”RR‹RdR‘(((sGlib/python2.7/site-packages/sympy/functions/combinatorial/factorials.pyR‰^s55     RŒcB s€eZdZdd„Zed„ƒZed„ƒZd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „ZRS( sBImplementation of the binomial coefficient. It can be defined in two ways depending on its desired interpretation: .. math:: \binom{n}{k} = \frac{n!}{k!(n-k)!}\ \text{or}\ \binom{n}{k} = \frac{ff(n, k)}{k!} First, in a strict combinatorial sense it defines the number of ways we can choose `k` elements from a set of `n` elements. In this case both arguments are nonnegative integers and binomial is computed using an efficient algorithm based on prime factorization. The other definition is generalization for arbitrary `n`, however `k` must also be nonnegative. This case is very useful when evaluating summations. For the sake of convenience for negative integer `k` this function will return zero no matter what valued is the other argument. To expand the binomial when `n` is a symbol, use either ``expand_func()`` or ``expand(func=True)``. The former will keep the polynomial in factored form while the latter will expand the polynomial itself. See examples for details. Examples ======== >>> from sympy import Symbol, Rational, binomial, expand_func >>> n = Symbol('n', integer=True, positive=True) >>> binomial(15, 8) 6435 >>> binomial(n, -1) 0 Rows of Pascal's triangle can be generated with the binomial function: >>> for N in range(8): ... print([binomial(N, i) for i in range(N + 1)]) ... [1] [1, 1] [1, 2, 1] [1, 3, 3, 1] [1, 4, 6, 4, 1] [1, 5, 10, 10, 5, 1] [1, 6, 15, 20, 15, 6, 1] [1, 7, 21, 35, 35, 21, 7, 1] As can a given diagonal, e.g. the 4th diagonal: >>> N = -4 >>> [binomial(N, i) for i in range(1 - N)] [1, -4, 10, -20, 35] >>> binomial(Rational(5, 4), 3) -5/128 >>> binomial(Rational(-5, 4), 3) -195/128 >>> binomial(n, 3) binomial(n, 3) >>> binomial(n, 3).expand(func=True) n**3/6 - n**2/2 + n/3 >>> expand_func(binomial(n, 3)) n*(n - 2)*(n - 1)/6 References ========== .. 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